Square Root of 7500

The square root is the inverse of the square of the number. 7500 is not a perfect square. The square root of 7500 is expressed in both radical and exponential forms. In the radical form, it is expressed as √7500, whereas (7500)^(1/2) in the exponential form. √7500 ≈ 86.6025, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where long-division and approximation methods are used. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 7500 is broken down into its prime factors.

Step 1: Finding the prime factors of 7500. Breaking it down, we get 2 x 2 x 3 x 5 x 5 x 5 x 5: 2^2 x 3^1 x 5^4.

Step 2: Now we found out the prime factors of 7500. The second step is to make pairs of those prime factors. Since 7500 is not a perfect square, therefore the digits of the number can’t be grouped into perfect pairs. Therefore, calculating 7500 using prime factorization involves approximating the root.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step.

Step 1: To begin with, we need to group the numbers from right to left. In the case of 7500, we need to group it as 75 and 00.

Step 2: Now we need to find n whose square is less than or equal to 75. We can say n is ‘8’ because 8 x 8 = 64, which is less than 75. Now the quotient is 8, after subtracting 75 - 64, the remainder is 11.

Step 3: Now let us bring down 00, making the new dividend 1100. Add the old divisor with the same number 8 + 8 = 16, which will be our new divisor prefix.

Step 4: The new divisor will be 16n, where we need to find the value of n such that 16n x n ≤ 1100. Let n = 6, then 166 x 6 = 996.

Step 5: Subtract 996 from 1100, the difference is 104, and the quotient is 86.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 10400.

Step 7: Repeat the process to refine the accuracy. For instance, choose n to refine the divisor to 1732, and continue the process until the desired precision is achieved. So, the square root of √7500 is approximately 86.60.

The approximation method is another method for finding square roots, and it is an easy method to find the square root of a given number. Now let us learn how to find the square root of 7500 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √7500. The smallest perfect square less than 7500 is 6400 (80^2) and the largest perfect square greater than 7500 is 8100 (90^2). √7500 falls somewhere between 80 and 90.

Step 2: Now we need to apply a linear approximation formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square) Going by the formula (7500 - 6400) ÷ (8100 - 6400) = 0.6471. Using the formula, we identified the approximate decimal point of our square root. The next step is adding the value we got initially to the decimal number which is 80 + 0.6471 = 86.6471, so the square root of 7500 is approximately 86.60.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16 and the inverse of the square is the square root, that is √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
   
• Principal square root: A number has both positive and negative square roots; however, it is usually the positive square root that has more prominence due to its uses in the real world. That is why it is also known as the principal square root.
   
• Factorization: The process of breaking down a number into the product of its prime factors.
   
• Long division method: A method used to find the square root of non-perfect squares, especially useful for larger numbers.